# Effect of Wall Flexibility on the Deformation during Flow in a Stenosed Coronary Artery

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Flow Domain and Artery Wall

#### 2.2. Meshing and Grid Independence

#### 2.3. Flow Modeling

#### 2.4. Fluid-Structure Interaction

#### 2.5. Structural Modeling

#### 2.6. Implementation of the Holzapfel Model

## 3. Results

#### 3.1. Steady State Flow

#### 3.2. Wall Deformation in Steady Flow

#### 3.3. Wall Deformation in Pulsatile Flow

## 4. Discussion

- Maximum wall deformation predicted by the physiologically accurate Holzapfel model is ≈50% that predicted by the Neo-Hookean model.
- Maximum wall deformation predicted is highest for the Linear-Elastic model.
- Wall deformation of both Neo-Hookean and Mooney-Rivlin model are consistently lower (5.6% and 66.7%, respectively) than that of the Linear-Elastic model (for the same shear modulus).
- Maximum wall deformation occurs well before the peak stenosis location: 6mm from the entrance, compared to 8.3 mm where the peak stenosis is located.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

FSI | Fluid-Structure Interaction |

## References

- Zarins, C.K.; Giddens, D.P.; Bharadvaj, B.K.; Sottiurai, V.S.; Mabon, R.F.; Glagov, S. Carotid bifurcation atherosclerosis. Quantitative correlation of plaque localization with flow velocity profiles and wall shear stress. Circ. Res.
**1983**, 53, 502–514. [Google Scholar] [CrossRef] [PubMed] - Ku, D.N.; Giddens, D.P.; Zarins, C.K.; Glagov, S. Pulsatile flow and atherosclerosis in the human carotid bifurcation. Positive correlation between plaque location and low oscillating shear stress. Arterioscler. Thromb. Vasc. Biol.
**1985**, 5, 293–302. [Google Scholar] [CrossRef] - Ku, D.N. Blood flow in arteries. Annu. Rev. Fluid Mech.
**1997**, 29, 399–434. [Google Scholar] [CrossRef] - Berger, S.A.; Jou, L.D. Flows in stenotic vessels. Annu. Rev. Fluid Mech.
**2000**, 32, 347–384. [Google Scholar] [CrossRef] - Holzapfel, G.A.; Ogden, R.W. Constitutive modelling of arteries. Proc. R. Soc. A
**2010**, 466, 1551–1597. [Google Scholar] [CrossRef] - Tang, D.; Yang, C.; Kobayashi, S.; Ku, D.N. Generalized finite difference method for 3-D viscous flow in stenotic tubes with large wall deformation and collapse. Appl. Numer. Math.
**2001**, 38, 49–68. [Google Scholar] [CrossRef] - Lee, N.; Xu, T. Modelling of flow and wall behaviour in a mildly stenosed tube. Med. Eng. Phys.
**2002**, 24, 575–586. [Google Scholar] [CrossRef] - Cheema, T.A.; Park, C.W. Numerical investigation of hyperelastic wall deformation characteristics in a micro-scale stenotic blood vessel. Korea-Aust. Rheol. J.
**2013**, 25, 121–127. [Google Scholar] [CrossRef] - Das, A.; Paul, A.; Taylor, M.D.; Banerjee, R.K. Pulsatile arterial wall-blood flow interaction with wall pre-stress computed using an inverse algorithm. Biomed. Eng. Online
**2015**, 14, S1–S18. [Google Scholar] [CrossRef] [PubMed] - Tang, D.; Yang, C.; Zheng, J.; Woodard, P.K.; Saffitz, J.E.; Sicard, G.A.; Pilgram, T.K.; Yuan, C. Quantifying effects of plaque structure and material properties on stress distributions in human atherosclerotic plaques using 3D FSI models. J. Biomech. Eng.
**2005**, 127, 1185–1194. [Google Scholar] [CrossRef] [PubMed] - Krishnakumar, R.; Balakrishnan, K.R. Influence of lumen shape and vessel geometry on plaque stresses: possible role in the increased vulnerability of a remodelled vessel and the “shoulder” of a plaque. Heart
**2005**, 91, 1459–1465. [Google Scholar] [CrossRef] [PubMed] - Holzapfel, G.A.; Gasser, T.C.; Ogden, R.W. A New Constitutive Framework for arterial wall mechanics and a comparative study of material models. J. Elast.
**2000**, 61, 1–48. [Google Scholar] [CrossRef] - Baghel, A.K.; Naik, S.; Rajagopal, A.; Anand, M. Simulation of blood flow in the stenosed left coronary artery. In Proceedings of the International Conference on Computational Systems in Engineering and Technology, Chennai, India, 7–8 March 2014. [Google Scholar]
- Cho, Y.I.; Kensey, K.R. Effects of the non-newtonian viscosity of blood on flows in a diseased arterial vessel. Part I: Steady flows. Biorheology
**1991**, 28, 241–262. [Google Scholar] [PubMed] - Nichols, W.W.; O’Rourke, M.F.; Vlachopoulos, C. McDonald’s Blood Flow in Arteries: Theoretical, Experimental, and Clinical Principles, 6th ed.; CRC Press: Boca Raton, FL, USA, 2011; p. 39. [Google Scholar]
- Schultze-Jena, B.S. Über die schraubenförmige Struktur der Arterienwand. Gegenbauers Morphol. Jahrb.
**1939**, 83, 230–246. [Google Scholar] - Staubesand, J. Anatomie der Blutgefäße. I. Funktionelle Morphologie der Arterien, Venen und arterio-venösen Anastomosen. In Angiology; Ratschow, M., Ed.; Thieme: Stuttgart, Germany, 1959; pp. 23–82. [Google Scholar]
- ANSYS Inc. A Large Deformation Neo-Hookean User Material in ANSYS; ANSYS Inc.: Canonsburg, PA , USA, 2008. [Google Scholar]
- Nolan, D.R.; Gower, A.L.; Destrade, M.; Ogden, R.W.; McGarry, J.P. A robust anisotropic hyperelastic formulation for the modelling of soft tissue. J. Mech. Behav. Biomed. Mater.
**2014**, 39, 48–60. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**Flow domain and wall geometry (

**a**), along with meshing (

**b**), recreated from angiogram of the left coronary artery.

**Figure 5.**Wall deformation along artery length during steady flow of blood obtained for Neo-Hookean model and Holzapfel model.

**Figure 6.**Wall deformation of artery ([5] model vs Neo-Hookean model) along the artery length.

**Figure 7.**Variation of artery wall deformation at location of maximum stenosis during pulsatile flow. Wall is modeled as Holzapfel material.

Model | Parameter (s) | Value (s) |
---|---|---|

Rigid | $\mu $ | ∞ |

Linear-Elastic | $\mu $, $\nu $ | 10.346 kPa, 0.45 |

Neo-Hookean | $\mu $ | 10.346 kPa |

Mooney-Rivlin | ${\mu}_{1},{\mu}_{2}$ | 10.346 kPa, 3.0 kPa |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Kallekar, L.; Viswanath, C.; Anand, M.
Effect of Wall Flexibility on the Deformation during Flow in a Stenosed Coronary Artery. *Fluids* **2017**, *2*, 16.
https://doi.org/10.3390/fluids2020016

**AMA Style**

Kallekar L, Viswanath C, Anand M.
Effect of Wall Flexibility on the Deformation during Flow in a Stenosed Coronary Artery. *Fluids*. 2017; 2(2):16.
https://doi.org/10.3390/fluids2020016

**Chicago/Turabian Style**

Kallekar, Laxman, Chinthapenta Viswanath, and Mohan Anand.
2017. "Effect of Wall Flexibility on the Deformation during Flow in a Stenosed Coronary Artery" *Fluids* 2, no. 2: 16.
https://doi.org/10.3390/fluids2020016